Before considering doing it this way, please see Why is using a shell loop to process text considered bad practice?
Nevertheless, your attempt is close (one obvious issue is the whitespace around the = in your total_length assignment).
#!/bin/bash
sum=0
while IFS=, read -a fields; do
if [[ ${fields[12]} = "$1" ]]; then
(( sum += fields[5] ))
fi
done < file.csv
printf 'sum: %d\n' "$sum"
You can shorten the if..then..fi using short-scircuit logic if you prefer ex.
[[ ${fields[12]} = "$1" ]] && (( sum += fields[5] ))
There's no need to skip the header line explicitly since it will fail the ${fields[12]} = "$1" test.